How this instrument works
A geometric sequence is a list built by repeating the same multiplication: start at a₁ and scale by the common ratio r each step. Because every step multiplies by exactly r, reaching the nth term takes n − 1 multiplications from the start, which is why the formula reads aₙ = a₁ · rⁿ⁻¹ rather than a₁ · rⁿ — that missing final step is the single most common slip when students first apply it, the multiplicative twin of the off-by-one mistake arithmetic sequences invite with (n − 1)d.
The name 'geometric' is not decoration. Pick any interior term of the sequence and it equals the geometric mean of the two terms flanking it: aₙ² = aₙ₋₁ · aₙ₊₁. Take 2, 6, 18, 54, 162 — the middle term 18 squared is 324, and 6 × 54 is also 324. That identity is what separates a geometric sequence, an entire progression of terms, from a geometric mean, which is just one number summarizing a pair; the sequence produces infinitely many such means along its own length, one for every interior position.
The ratio decides everything about the sequence's shape. Set r above 1 and the terms grow without bound; set r between 0 and 1 and they shrink toward zero without ever reaching it, since a positive number times a positive fraction stays positive; set r to exactly 1 and the sequence goes flat, repeating a₁ forever; set r negative and the sign flips every step. A ratio of 0 is the one degenerate case — every term after the first collapses to zero, and the progression stops behaving like one.
- Enter the sequence's starting value in First term, a₁ — the number the whole progression is built from.
- Enter the fixed multiplier in Common ratio, r; use a value between 0 and 1 for a shrinking sequence, or a negative value for one that flips sign each step.
- Set Term number, n to the position you want to inspect — n = 1 returns the first term itself, before any multiplication happens.
- Read aₙ for the value sitting at that position, computed as a₁ times r raised to the n − 1 power.
Worked example — starting at 2, tripling each step
Take a₁ = 2, r = 3, and n = 5. The fifth term is aₙ = 2 × 3⁵⁻¹ = 2 × 3⁴ = 2 × 81 = 162. Counting the terms out directly confirms it: 2, 6, 18, 54, 162 — five terms, each three times the one before, landing on 162 at the fifth position, exactly where the formula puts it.
The interior-term check works here too. Look at the third term, 18: squaring it gives 324, and multiplying its neighbours, 6 × 54, also gives 324. That match is not a coincidence — it holds for every interior term of every geometric sequence, and it is exactly why sequences built this way earned the name 'geometric' in the first place.
Questions
What is the formula for a geometric sequence?
aₙ = a₁ · rⁿ⁻¹, where a₁ is the first term, r is the common ratio, and n is the position you want. The exponent is n − 1, not n, because reaching the nth term takes n − 1 multiplications by r starting from the first term itself, which already occupies position one before any ratio is applied.
How is a geometric sequence different from an arithmetic one?
An arithmetic sequence adds a fixed amount each step, aₙ = a₁ + (n − 1)d; a geometric sequence multiplies by a fixed ratio instead, aₙ = a₁ · rⁿ⁻¹. Constant differences between consecutive terms signal arithmetic; constant ratios signal geometric — and because multiplication compounds, a geometric sequence with r above 1 eventually overtakes any arithmetic sequence no matter how large d is.
Is a geometric sequence the same thing as a geometric mean?
No. A geometric mean is a single number summarizing two values, √(xy), while a geometric sequence is an entire list of terms produced by repeated multiplication. They connect closely — every interior term of a geometric sequence equals the geometric mean of its two neighbours — but generating a whole sequence is a different task from computing one mean.
What happens when the common ratio is 1, 0, or negative?
A ratio of exactly 1 leaves every term identical to a₁, so the sequence is flat. A ratio of 0 sends every term after the first to zero. A negative ratio flips the sign each step, so the sequence alternates between positive and negative while its size still grows or shrinks depending on whether the ratio's absolute value sits above or below 1.
Why does a ratio between 0 and 1 make the sequence shrink toward zero?
Multiplying by a number smaller than 1 always produces something smaller than what you started with, so each new term is a fraction of the one before it — 100, 50, 25, 12.5 for a ratio of 0.5. The terms keep approaching zero but, for any finite term number, never actually land on it, since a positive number times a positive fraction stays positive.
Can the term number or common ratio be negative or a fraction?
The common ratio can be almost any real number — negative values and fractions both produce valid, well-defined sequences, alternating or shrinking as described above. The term number n should stay a positive whole number, since it counts a specific position in the list; the formula still computes for a non-integer n, but the result no longer corresponds to an actual term.